Verify the Identity (cos(x))/(1-sin(x))=(1+sin(x))/(cos(x))

Problem

cos(x)/(1−sin(x))=(1+sin(x))/cos(x)

Solution

1. Identify the left side of the identity to be transformed.

cos(x)/(1−sin(x))

2. Multiply the numerator and the denominator by the conjugate of the denominator, which is 1+sin(x)

(cos(x)*(1+sin(x)))/((1−sin(x))*(1+sin(x)))

3. Expand the denominator using the difference of squares formula (a−b)*(a+b)=a^2−b^2

(cos(x)*(1+sin(x)))/(1−sin^2(x))

4. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 which implies 1−sin^2(x)=cos^2(x)

(cos(x)*(1+sin(x)))/cos^2(x)

5. Simplify the fraction by canceling a factor of cos(x) from both the numerator and the denominator.

(1+sin(x))/cos(x)

Final Answer

cos(x)/(1−sin(x))=(1+sin(x))/cos(x)

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